ODE
\[ 2 a^2 y(x)+2 \left (1-x^2\right ) y''(x)-x y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.176092 (sec), leaf count = 66
\[\left \{\left \{y(x)\to \left (x^2-1\right )^{3/8} \left (c_1 P_{\frac {1}{4} \left (\sqrt {16 a^2+1}-2\right )}^{\frac {3}{4}}(x)+c_2 Q_{\frac {1}{4} \left (\sqrt {16 a^2+1}-2\right )}^{\frac {3}{4}}(x)\right )\right \}\right \}\]
Maple ✓
cpu = 0.283 (sec), leaf count = 55
\[\left [y \left (x \right ) = \textit {\_C1} \left (x^{2}-1\right )^{\frac {3}{8}} \LegendreP \left (\frac {\sqrt {16 a^{2}+1}}{4}-\frac {1}{2}, \frac {3}{4}, x\right )+\textit {\_C2} \left (x^{2}-1\right )^{\frac {3}{8}} \LegendreQ \left (\frac {\sqrt {16 a^{2}+1}}{4}-\frac {1}{2}, \frac {3}{4}, x\right )\right ]\] Mathematica raw input
DSolve[2*a^2*y[x] - x*y'[x] + 2*(1 - x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (-1 + x^2)^(3/8)*(C[1]*LegendreP[(-2 + Sqrt[1 + 16*a^2])/4, 3/4, x] + C[2]*LegendreQ[(-2 + Sqrt[1 + 16*a^2])/4, 3/4, x])}}
Maple raw input
dsolve(2*(-x^2+1)*diff(diff(y(x),x),x)-x*diff(y(x),x)+2*a^2*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*(x^2-1)^(3/8)*LegendreP(1/4*(16*a^2+1)^(1/2)-1/2,3/4,x)+_C2*(x^2-1)^(3/8)*LegendreQ(1/4*(16*a^2+1)^(1/2)-1/2,3/4,x)]